Final answer:
To verify the identity algebraically, we can simplify both sides of the equation and show that they are equal. Starting with the left side, we can use trigonometric identities to rewrite and simplify the expression until it matches the right side of the equation. In this case, the simplification leads to both sides being equal to 6, confirming the identity.
Step-by-step explanation:
To verify the identity algebraically, we will simplify both sides of the equation and show that they are equal. Starting with the left side:
6 cot²(y)(sec²(y) − 1) = 6
Using the identities cot²(y) = 1 + tan²(y) and sec²(y) = 1 + tan²(y), we can rewrite the equation as:
6(1 + tan²(y))(1 + tan²(y) - 1) = 6
Simplifying further:
6(1 + tan²(y))(tan²(y)) = 6
Expanding and simplifying:
6tan²(y) + 6tan⁴(y) = 6
Since tan⁴(y) + tan²(y) = 1, we can rewrite the equation as:
6tan²(y) + 6(1 - tan²(y)) = 6
Simplifying further:
6tan²(y) + 6 - 6tan²(y) = 6
Combining like terms:
6 = 6
Since both sides of the equation are equal, the identity is verified.